The Yellowstone permutation: limsup R(t)/t <= 0.68605, liminf n_p/p > 1.4576, and Conjecture 7 for large n

This manuscript continues the series on the Yellowstone permutation a (OEIS A098550). Let R(t) be the least odd prime with no multiple among a(1), ..., a(t), for a prime p let n_p be the first index with p | a(n_p), let M(n) = maxj<=n a(j), and let E(t) be the number of even terms among a(1), ..., a(t). The previous manuscript of the series proved limsup R(t)/t <= 0.9265. We report the program's present asymptotic theorem: limsup R(t)/t <= 0.68605 (the worst class value is exactly 13721/20000, so the bound is not strict), liminf n_p/p >= 1/0.68605 > 1.4576, M(n) <= (0.68605 + o(1)) n ln n, and liminf E(t)/t >= 0.2695. Hence the inequality a(n) <= (pi/2) n ln n of OEIS Conjecture 7 holds for all n >= n_0, where n_0 is ineffective. The proof rests on the published manuscripts cited below and on a chain of unpublished program notes, each proved and refereed at least twice by AI referees, and is given in this manuscript as statements with proof sketches and references to those notes. The chain consists of the notes treated in the previous manuscript (gap (ii), the parity floor and the hole sieve), a hole law refined in six notes, a numerical refinement of the odd-mask chain, a pair law for the masks above dense holes (constant 0.6933), a partner law for first-exit masks together with half classes at the top (0.68915; most of that gain comes from the half classes), quarter and eighth classes at the top with a start lemma (0.6874; a refinement of the class cover, no new law), and an X-start hole law (0.68605): the hole recurrence is read as a recurrence for the number of used evens above 2ut, started at the upper end of the class, which damps the class-width term of the odd-mask comparison at the top classes. The constants rest on optimiser runs (double precision) reproduced by AI numerics referees; the worst class value was checked by an exact-rational verifier. The manuscript also records where this method stops: the extrapolated width-zero limit, about 0.6854, of the class refinement is a heuristic, not a bound, and, by an unrefereed reading (not a theorem), the odd-mask start level of about 1.37 is not moved by this family of recurrences. A computation of a(1..8.3*10^10) by two independent programs (a run certificate, not a proof) is quoted as a consistency check only; it does not reach n_0. All thresholds are ineffective and the results are purely asymptotic. Conjecture 7 for all n is NOT proved, no explicit n_0 is given, and the program's conditional reductions of Conjecture 7 for all n are not part of this manuscript. AI-assistance and review status: The mathematical content was produced by AI agents (Anthropic Claude) in the author's research program on OEIS A098550, and the text was written by AI writer and fixer agents from the program's internal notes, synthesis and referee records. Each program note used is refereed at least twice by independent AI referee agents (separate Claude instances that had not written the text under review): for the hole-law notes and the numerical refinement one proof referee and one numerics referee; for the pair law, the partner law and the X-start hole law two proof referees and one numerics referee; for the quarter-class step two referees each covering its start lemma and its numerics. The assembled version 1.0 was checked as a whole by two further AI referees, REF-614 (mathematical fidelity against the source notes and runs; CORRECT WITH FIXES, no mathematical defect, no constant changed) and REF-615 (citations, status claims and wording; READY WITH FIXES); all their fixes were applied by an AI fixer agent and have not been re-refereed. Details are in the title-page footnote and Section 6 (Honest status) of the manuscript. All referee reports cited are AI-generated. No part has been reviewed by a human mathematician. Dependencies: Part I (doi:10.5281/zenodo.23437783), Part II (doi:10.5281/zenodo.23449375), Part III (doi:10.5281/zenodo.23557280), the manuscript on the linear prime frontier (doi:10.5281/zenodo.23558450) and the manuscript on limsup R(t)/t < 1 (doi:10.5281/zenodo.23604646), and classical results (prime number theorem, Mertens' theorem, Legendre's count, Rankin's trick; Robin's bound, Rosser-Schoenfeld and Selberg's sieve through the cited manuscripts). The unpublished program notes (GIIRICH, GIIPUSH, GIIONE, GIITWO, GIIHOLE, GIIODD, GIIEVEN, GIITOP, GIIMID, GIIALL, GIIBTOP, GIIRMASK, GIIFEXIT, GIIQTR, GIIHLAM, and the synthesis GIISYN) are internal and are not published with this manuscript; the first four are treated in the manuscript doi:10.5281/zenodo.23604646.